Now, let us compare these results with Maker-Breaker games in which both players play randomly. Then, having bias \((m:b)\) , Maker's final graph behaves similarly to a random graph \(G \sim G_{n,p}\)  with \(p= m/(m+b)\) . It is well known that the (sharp) threshold probability \(p^*\)  for \(G \sim G_{n,p}\)  being connected or containing a Hamilton cycle is of size \((1+o(1))\ln n / n\)  (see e.g. [1]}, [2]}). If \(m=1\)  this corresponds to \(b = (1 + o(1))n / \ln n\) , which matches the above mentioned threshold biases perfectly. Thus, for almost all values of \(b\) , the outcome of the \((1:b)\)  Maker-Breaker connectivity game and Hamiltonicity game on \(K_n\)  is most likely the same, no matter if both players play perfectly or just pick their edges at random. Usually this phenomenon is referred to as the probabilistic intuition. There is a huge variety of games which fulfil this intuition, for example the perfect matching game, the Hamiltonicity game [3]}, and the \((m:b)\)  connectivity game if \(m = o(\ln n)\)  [4]}, but there also exist games for which this intuition fails, such as the \(H\) -game [5]},
the \(K_t\) -factor game [6]} or the diameter game [7]}.
